Question: All of the 4th grade teachers and students from Gardner Bullis went on a field trip to an art museum. Tickets were $$6.00$ each for teachers and $$2.50$ each for students, and the group paid $$40.50$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$18.00$ each for teachers and $$12.50$ each for students, and the group paid $$166.50$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6x+2.5y = 40.5}$ ${18x+12.5y = 166.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-18x-7.5y = -121.5}$ ${18x+12.5y = 166.5}$ Add the top and bottom equations together. $ 5y = 45 $ $ y = \dfrac{45}{5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {6x+2.5y = 40.5}$ to find $x$ ${6x + 2.5}{(9)}{= 40.5}$ $6x+22.5 = 40.5$ $6x = 18$ $x = \dfrac{18}{6}$ ${x = 3}$ You can also plug ${y = 9}$ into $ {18x+12.5y = 166.5}$ and get the same answer for $x$ ${18x + 12.5}{(9)}{= 166.5}$ ${x = 3}$ There were $3$ teachers and $9$ students on the field trips.